Thermal properties of stripe-geometry laser diodes

Optica Applicata, Vol. X I I I , No. 3, 1983

Thermal properties o f stripe-geometry laser diodes

Wł o d z im ie r z Na k w a s k i

Institute of Physics, Technical University of Łódź, ul. Wólczańska 219, 93-005 Łódź,

Poland-In this work thermal properties of stripe-geometry double-heterostructure GaAs- (AlGa)As laser diodes have been analyzed. The space transformation reducing the non­ linear thermal conduction equation to the linear one has been used. The transfor­ mation has replaced the nonhomogeneous GaAs-(AlGa)As double-heterostructure of the laser diode for the homogeneous, GaAs or (AlGa)As homostructure.

In calculations of the heat sources distribution, nonradiative recombination,. reabsorption of radiation, Joule heating, the radiative transfer of the spontaneous radiation through the passive layers as well as the current spreading effect have been taken into account.

1 . Introduction

In recent years an intense effort has been devoted to the development of laser diodes, the simplest and most compact lasing devices. Most of them utilize a stripe-geometry. This development has been stimulated by the optical telecom­ munication where a stripe-geometry laser diode [1-8] is used as a carrier wave generator assuring an efficient coupling with a fiber lightguide [9-11].

Due to the operation of the laser diode the temperature increases within its volume. This is induced mainly by nonradiative recombination as well as. reabsorption of radiation and the Joule.heating. This rise of temperature deterio­ rates the laser diode performance causing an increase in the threshold current density, a decrease in the output power and the external quantum efficiency,, as well as a shift of the whole spontaneous radiation spectrum and the mode peaks on spectral characteristics. In the case of the stripe-geometry laser diodes without a built-in waveguide, the temperature distribution in the p-n junction plane improves the stabilization of the radiation filament.

Because of all the above mentioned effects, the operational characteristics of laser diodes are influenced by their thermal properties which have been investigated by many authors. The one-dimensional heat extraction in the oxide-insulated Stripe laser diodes was described by Ga r e l- Jo nes and Dy m e n t

[12]. The two-dimensional case for the stripe-geometry laser diodes without, oxide barriers (e.g., proton bombardment stripe laser diodes or planar stripe laser diodes) was analysed by Joyce and Dix o n [13]. Subsequent papers

devoted to this subject added some improvements to the work of Joyce and Dixon. Newman et al. [14] took into consideration the radiative energy transfer of the spontaneous radiation. Dud A et al. [15] investigated the relative influence of various heat sources (including the Joule heating)

282 _{W . N}a k w a sk i

on the temperature distribution in the laser diodes. Buus [16, 17] took ac­ count of the current spreading effect. Ito and Kim uka [18] considered the heat

extraction from the top surface of the laser crystal (thermal radiation and con­ duction through a bonding -wire). St ev en to n et al. [19] and Yano et al. [20]

adapted this model to the InGaAsP stripe laser diodes.

The solution given by Joyce and Dixon has the form of an infinite series with unknown coefficients. These coefficients can be determined by solving a great number of equations for the continuity of temperature and heat flux on each boundary between layers. This is the main disadvantage of the above method. In this paper, the approximate, analytical solution of the thermal conduction equation in the active layer of the stripe-geometry laser diode is presented. In this case, the expansion coefficients depend only on the laser construction parameters and the energy source.

The basic assumptions of the model are presented in Sec. 2; the space trans­ formation which reduces the nonhomogeneous medium of the laser diode to homogeneous one is the subject of Sec. 3; the distribution of the heat sources is discussed in Sec. 4. The solution and results are presented in Sec. 5 and Sec. 6, respectively.

The present work is a continuation of our previous papers devoted to the thermal properties of broad contact laser diodes in the steady-state condition [21-23] and in the transient-state condition [24-29] as well as to the thermal properties of stripe-geometry laser diodes [30].

2 . Assumptions

The standard construction of the stripe-geometry double-heterostructure GaAs-(AlGa)As laser diode without oxide barriers is shown in Fig. 1, where

Pig. 1. Standard construction of stri­

pe-geometry, double-lieterostructure

GaAs-(AlGa)As laser diode without oxide harriers (A - the active region)

the thermal conductivities are taken from papers [31-33]. Standard parameters of the laser used in calculations are given in Table 1.

Thermal properties of stripe-geometry laser diodes 283

T a b l e 1. The param eters of standard stripe-geom etry double-heterostructure G aAs-(AlG a)A s laser diode P ara­

m eter Value U nit Param eter V alue U n it

W 300 jim _Vbp 0.55 — 8 15 jim _Vext 0.3 — -L 400 jim _m 1 — I 150 mA _6s 4.26· 10“ 4 i l in ith 100 m A _es 7.00· 10~s ftm V 1.7 V _6l 1.40· 10~4 ilm

In this case, because of the position dependence of the thermal conductivity X, the thermal conduction equation is nonlinear:

V(AVT) = ~ g (1)

■where T is temperature and g is power density of heat sources. The boundary conditions may be formulated by assuming: i) a negligible heat extraction from the top and side walls

= 0, (2)

X—±WJ2

= 0, (3)

y=0

ii) a symmetry of the construction according to the y-axis

= 0, (4)

x=0

iii) an infinite thermal capacity of the heat sink

T (y = yHS) = TA (5)

where yHS denotes points on the external surface of the heat sink and T A is ambient temperature.

The main heat source in the active layer is placed very close to the heat sink. High efficiency of the heat extraction process in the laser diode seemingly enables us to formulate the boundary conditions (2) and (3) in a simpler form: T[x = ± ( W/2)) = T (y = 0) = 0. The measurements [34-36] of the tem­ perature distributions on the mirror facet have, however, proved the inade­ quacy of the above mentioned assumption.

8T 8x 8T 8x 8T 8y

W. Naxwaski

284

3 . Transfonnation

The space transformation presented in papers [37] enables, for the one-dimen­ sional case, the substitution of the thermally equivalent GaAs homostructure for the GaAs-(AlGa)As double-heterostructure. In this way, the thermal conduction equation (2) becomes linear:

V2T (* ,y ) 9 (x ,y )

X ' (6)

The transformation reduces the multi-layer, nonhomogeneous structure of the double-heterostructure (PH) GaAs-(AlGa)As laser diode to the homo­ structure GaAs laser diode replacing each i-th layer of thickness t{ and ther­ mally conductivity with the thermal equivalent GaAs layer of thickness <'· and thermal conductivity

(7)

Similar transformation for the two-dimensional case under consideration involves an alteration Of the lateral spreading of the heat flux; In this way, the transformation into the GaAs homostructure lowers the temperature in the active region in comparison with the real distribution, and the analogous transformation into the (AlGa)As homostructure raises this temperature. In the next Sections we shall try to compare the real temperature distribution in the p-n junction plane with the arithmetic mean of the results of both the transformations.

The transformation is performed for all the layers. In the case of the heat sink, the method proposed by La f f et al. [38] is used

(8)

where - effective width of^the heat flux flowing into the heat sink, P - dis­ sipated power, L - length of the laser resonator, Tav - averaged temperature of the active region, and At - thickness of a hypothetic layer of thermal con­ ductivity Xt inserted between the laser chip and the heat sink. The thermal resistance QHS of the copper heat sink is then equal to

^ 1 , / 4L \

Q* s ~ 2 ^ i ln (“s 7 ) (9)

where X12 - thermal conductivity of copper.

The increase in temperature in the heat sink may be presented as follows: A ^hs — Qiisit S = - j —t

Thermal properties of stripe-geometry laser diodes 285

where the averaged density qt of the total heat flux flowing into the heat sink is

(11)

and t12 is the thickness of the copper layer of the thermal resistance QHS. Then the thickness t12 of the thermally-equivalent GaAs layer will be

All thicknesses of the layer before an4 after the transformation into GaAs, as well as the transformation into (AlGa)As are listed in Table 2.

T a b le 2. The space transform ation coefficients and th e th ick ­ nesses o f layers o f th e standard stripe-geom etry laser diode before and after th e transform ation

Layer num ber Space transform ation coefficient Thickness [pm ] before after transform ation 1 0.0853 0.2 0.02 2 0.272 91.8 24.97 3 0.272 1 0.27 4 1 3 3 . 5 r. 0.272 0.2 0.05 6 1 2 2 7 0.272 2 0.54 8 0.582 0.1 0.06 9 0.175 0.15 0.03 10 0.0403 1.2 0.05 11 0.147 2 0.29 12 0.032 OO 0.99

4 . Heat sources

In the laser diode volume there exist three sorts of the heat sources :

i) The main heat source is placed in the active region and results from non- radiative recombination and reabsorption of the generated radiation. Its power density is [35]

9a = ~ - / % P) + Ü - it h ) [1 - Vcxt - (1 ~ Vi) %P/ ] } (13)

h

where U is the voltage drop at the p-n junction, j and j th are the supply current density and threshold current density, respectively, and ts is the thickness

286 W;> Na k w a s e i

of the active layer, %p, rjcxt and % are the internal quantum efficiency of the spontaneous emission, the external differential quantum efficiency of the lasing and the internal quantum efficiency of the lasing , respectively. Coefficient / describes the radiative transfer of the spontaneous radiation generated in the active region through the wide-gap, passive (AlGra)As layers. It may be calculated in the following;"way [39]:

/ 2 sin2 arc sin | l — 0.62 ~ ~ (14)

where nR - refractive index of the active region material and Ax 4T, - difference in AlAs content between passive and active layers.

ii) In the GaAs capping layer and in the top layer of the substrate (layer 3 in Fig. 1) there exist the additional heat sources that are connected with the absorption of the spontaneous radiation transferred radiatively through the passive layers. Assuming, the homogeneous distribution of the power density in both layers, they may be represented as follows:

02V, 7 — ^Jth ^?sp / 2t7 ’ 02V, 3 — U jth Vbp f 2t3 ' (15) (16) lii) A supplementary influence is exercised by the Joule heating generated · in each layer with a density

Qj.i = 326i, i = 1 , 2 , 3 , . . . . , 1 2 (17)

where - electrical resistivity of i-th layer material.

In all the formulae given in this Section, the current spreading has not been taken into consideration. These formulae are adequate for laser diodes with a good current confinement. A more general example will be shown in Section 6.

The space transformation should be also performed for the power densities. This is due to the changes of the layer thicknesses. The transformation coefficient of the heat power densities is therefore the following:

fg.i A

9 i

i = 1 , 2 , 3 , 12. (18)

All the heat power densities, before and after the space transformations, are listed in Table 3. In the next Sections the primes will be omitted because only the values after the transformation will be considered.

T a b l e 3. H eat power d en sities before and after th e transform ation

H eat source L ocalization

D en sity [W m - 3 ] before after transform ation N otation for calculations

N onradiative recom bination and reabsorption o f radiation in th e a ctiv e layer

A ctiv e layer _9s 1.398· 1014 5 .1 4 1 -lO14 _9s

A bsorption o f spontaneous

radiation transferred radia- p-G aA s 9i,Tr 2.577· 1012 9 .4 6 3 -1012 _{9i = 9i>Tr + 9i,J}

tiv e ly through p assive layers w-GaAs(Tr) _9s 5.153· 1012 1.891-10“ _9s

w-GaAs _9s 3 .3 -107 * *

JV-(AlGa)As _9t 2.66· 1011 2.66· 10u _9s

P -(A lG a)A s _9s 8.75· 1010 8.75· 1010 _9s

p-G aAs _9i,J 4.38· 1010 1 .6 1 -1011 _9i = 9 i,tr + 9n,J

Jou le h eatin g Ti _9s 1.4-10» * *

P t _9» 2.9· 10» * *

A u _9io 8 -1 0 7 * *

In _9n 2-6 -1 0 » ♦ *

H eat sink _9is 5 -1 0 7 * *

* means that this heat source with a density < IQ1# Wm-3 will not he taken Into farther account

T h e rm a l p ro p e rt ie s o f st rip e -g e o m e tr y la se r d io d e s 2 8 7

5 . Solution

288 _{W. Na k w a sk i}

The problem considered reduces now to the solution of the linear thermal conduction equation

V2T (x , y) = —

with the boundary conditions: 8T 8x 8T X=0 8x 8T = 0, = 0 , 8y X=W/ 2 = 0 y=0 T (x = a12) = T A where

(Ji . for \x\ < £/2 a n d y e ( a ^ , a{y elsewhere

9(®,y) = {

q

i

ai — h t i 1 > % > 3, . ., 12 k=1

and g{ are given in Table 3.

The solution of this problem is the following:

T ( x ,y ) = T A 4 V | <?m«os(Lmy)

f

AWa12 E-J l

[2

71 — 0

XCOS_{( - * » +} sin(Kn(S/2))_{2 , l cos(Knx)}

E n(K 2n + L 2m) _]}■ with K n I 'm = 2nn W ’ (2m — l ) n 2 di o Gm = ^ 9 i i ^ { L mai) - s i n ^ u ^ ! ) ] . (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29)

Thermal properties of stripe-geometry laser diodes 289

6 . Results

Let us consider the arithmetic mean of both the solutions obtained with the aid of the transformations into GaAs (T J and (AlGa)As (T2) as a result of oixr calculations:

T = 4 - ( T ^ T J . (30)

For the standard stripe-geometry double-heterostructure GaAs-(AlGa)As laser diode (Fig. 1, Tab. 1) the new obtained solution (N) is compared with the exact solution (JD) of Joyce and Dixon in Fig. 2. It is evident that the observed discrepancy of both curves is insignificant.

F ig. 2. Tem perature d istrib ution s in th e p-n junction plane o f stripe- geom etry D E laser diode ; w ith

8 = 15 pm. JD - solu tion of Joyce and D ixon, N -n ew approxim ated solution, GaAs and AlGaAs. - solu ­ tion s for th e transform ations in to GaAs and (AlGa)As, resp ectively

In order to check this method, we perform an analogical comparison for the extremely different conditions:

i) 8 = 15 pm, but the thicknesses of the passive layers are f4 = 2 (tm and

<6 = 3 (xm,

ii) 8 == 10 ja.m., iii) 8 !== 5 pm..

Fig. 3. Temperature., d istri­ butions in th e p-n junction plane of strip e:geom etry B E

laser diode w ith <Sf = 15 pm and th e changed valu es -of-tbe layer thickn esses: <4 = 2 pm

and 3 pm

The results are shown in Figs. 3-5, respectively. I t turned out that the proposed method is relatively insensitive to the reasonable changes of the laser diode structure. The method may be used for the approximate determination of the temperature distribution in the p-n junction plane of the stripe-geometry laser diode.

290 W. Nakwaski

F ig. 4. Tem perature d istrib ution s in th e p -n junction p lane of strip e-geo­ m etry D E laser d iode w ith 8 = 10 (a

F ig. 5. Tem perature d istrib ution s in th e p-n junction p lane of strip e-geo­ m etry JDH laser diode w ith 8 = 5 jtm

In the case of stripe-geometry laser diodes without the effective confinement of the current path between the contact and the active regions (e.g., oxide insulated, proton bombarded and planar stripe lasers), the current spreading effect should be taken into consideration. This effect results from the finite electrical resistance of the p-layer and P-layer, and may be usually described in the following way [40-41]:

j(y ) = p i j l 1 + W - S I 2 y h

I

(31)

Thermal properties of stripe-geometry laser diodes 291 *0 = B jr 112- (33) n " l 2 n .kT \ 112 (34) . . ) -k i . □ II + «■· (35) j = I /L S , (36)

•with I - supply current, k - Boltzman constant, e - unit charge, nc - constant (for the D E GaAs-(AlGa)As laser diode: nc 2, [43]).

For the heat power densities, we assume the same relative lateral distribu- .tion as for the current density

j(y)·· 9t 0 j(y ) for x a{y , i = 3 , 4 , 5, 6, 7, elsewhere (37)

The solution of the thermal conduction equation (19) with g(x, y) given by (37) and the boundary conditions (20)-(23), is the following

oosKnx

TÏL

1

]|

I

no sint si(®) = - J X 00 Ci(a?) = - J t cost ■At n r sir

T +f —

292 W. Nakwasju The influence of the current spreading effect on the temperature distri­ bution in the D H stripe-geometry laser diodes for 8 = 15 {Jim and 8 — 1*0 fan is illustrated in Figs. 6 and 7, respectively. It is apparent that the temperature- induced waveguiding in the D H stripe-geometry laser diodes depends strongly on the efficiency of the current confinement, i.e., on the value of the com­ posite sheet resistance J2D.

F ig. Temperature d istri­ b utions in the p -n junction p la­ ne of stripe.geom etry J5H la ­ ser diode·w ith 8 '-= 15'.pm) apd various valu es o f sheet resi­ stance JBa

Burrs [16] has adapted the Joyce-Dixon model to the stripe-geometry D H laser with the current spreading effect. He obtained analogical results, but for points outside the active region the temperature distributions

depend-F ig. 7. Tem perature distributions in th e p-n

junction plane o f , stripe-geom etry DjBF laser diode w ith 8 = 10 fjtm and various valu e o f sh eet resistance ;

ed much more strongly on the sheet resistance l i n . This means that for this region the method described here is less exact. As far, however, as the electrical and optical properties of the laser diode are concerned, the temperature distribu­ tion outside the active region is much less important.

Thermal properties of stripe-geometry laser diodes 293

7. Conclusions

The approximate, analytical solution of the thermal conduction equation was obtained for the stripe-geometry double-heterostructure G-aAs-(AlGa)As laser diode. The space transformation reducing the nonhomogeneous medium of the laser diode to the homogeneous one was used. The results were compared with the exact, semi-analytical solution of Joyce and Dixon. The current spread­ ing effect was taken into consideration.

Acknowledgements - The author would like to thank Dr. J. Buus from the Electromagnetic Institute, Technical University of Denmark, Lynghy, for stimulating discussion on the ther­ mal problems in laser diodes, as well as Dr. E. Nilsson and the colleagues from the above Institute for the help in computer calculations.

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Beceived June 13, 1983 Термические свойства ленточных лазеров В настоящей работе произведен анализ термических свойств бигетеромуфтовых ленточных лазеров ОаАз-(АЮа) Ая. С этой целью применена трансформация пространства, редуцирующая нелинейное уравнение термической проводности к линейной форме. Эта трансформация заменяет неоднород­ ную бигетероструктуру ваА5-(АЮа)А5 муфтового лазера однородной гомоструктурой СаАэ или (АЮа)Аз. В расчетах распределения источников тепла учтены нелучистая рекомбинация, реабсорбция излучения, джоулево тепло, лучистый трансферт спонтанного излучения через пассивные спои, а также эффект токораспределения. Проверила М олгож ата Хейдрих